Remote Sensing of Environment 103 (2006) 27 – 42
www.elsevier.com/locate/rse
Reflectance quantities in optical remote sensing—definitions
and case studies
G. Schaepman-Strub a,b,⁎, M.E. Schaepman c , T.H. Painter d , S. Dangel b , J.V. Martonchik e
a
Nature Conservation and Plant Ecology Group, Wageningen University and Research Centre, Bornsesteeg 69, 6708 PD Wageningen, The Netherlands
b
RSL, Department of Geography, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
c
Centre for Geo-information, Wageningen University and Research Centre, 6700 AA Wageningen, The Netherlands
d
National Snow and Ice Data Center (NSIDC), University of Colorado at Boulder, 449 UCB, Boulder, CO 80309-0449, United States
e
Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, CA 91109, United States
Received 9 May 2005; received in revised form 1 March 2006; accepted 4 March 2006
Abstract
The remote sensing community puts major efforts into calibration and validation of sensors, measurements, and derived products to quantify and
reduce uncertainties. Given recent advances in instrument design, radiometric calibration, atmospheric correction, algorithm development, product
development, validation, and delivery, the lack of standardization of reflectance terminology and products becomes a considerable source of error.
This article provides full access to the basic concept and definitions of reflectance quantities, as given by Nicodemus et al. [Nicodemus, F.E.,
Richmond, J.C., Hsia, J.J., Ginsberg, I.W., and Limperis, T. (1977). Geometrical Considerations and Nomenclature for Reflectance. In: National
Bureau of Standards, US Department of Commerce, Washington, D.C. URL: http://physics.nist.gov/Divisions/Div844/facilities/specphoto/pdf/
geoConsid.pdf.] and Martonchik et al. [Martonchik, J.V., Bruegge, C.J., and Strahler, A. (2000). A review of reflectance nomenclature used in remote
sensing. Remote Sensing Reviews, 19, 9–20.]. Reflectance terms such as BRDF, HDRF, BRF, BHR, DHR, black-sky albedo, white-sky albedo, and
blue-sky albedo are defined, explained, and exemplified, while separating conceptual from measurable quantities. We use selected examples from the
peer-reviewed literature to demonstrate that very often the current use of reflectance terminology does not fulfill physical standards and can lead to
systematic errors. Secondly, the paper highlights the importance of a proper usage of definitions through quantitative comparison of different
reflectance products with special emphasis on wavelength dependent effects. Reflectance quantities acquired under hemispherical illumination
conditions (i.e., all outdoor measurements) depend not only on the scattering properties of the observed surface, but as well on atmospheric
conditions, the object's surroundings, and the topography, with distinct expression of these effects in different wavelengths. We exemplify differences
between the hemispherical and directional illumination quantities, based on observations (i.e., MISR), and on reflectance simulations of natural
surfaces (i.e., vegetation canopy and snow cover). In order to improve the current situation of frequent ambiguous usage of reflectance terms and
quantities, we suggest standardizing the terminology in reflectance product descriptions and that the community carefully utilizes the proposed
reflectance terminology in scientific publications.
© 2006 Elsevier Inc. All rights reserved.
Keywords: Reflectance; Terminology; Definition; Nomenclature; BRDF; Spectrodirectional; Vegetation; Snow
1. Introduction
Since its origins, the remote sensing community has
explored the angular distribution of reflectance through field
measurements, laboratory measurements, remotely sensed mea⁎ Corresponding author. Nature Conservation and Plant Ecology Group,
Wageningen University and Research Centre, Bornsesteeg 69, 6708 PD
Wageningen, The Netherlands.
E-mail address: [email protected] (G. Schaepman-Strub).
0034-4257/$ - see front matter © 2006 Elsevier Inc. All rights reserved.
doi:10.1016/j.rse.2006.03.002
surements, and modeling studies. Nicodemus et al. (1977)
defined a widely cited, unified approach to the specification of
reflectance and proposed a nomenclature to facilitate this
approach. In this work, they developed the theoretical framework for discussion of reflectance quantities, based on the
concept of the bidirectional reflectance distribution function
(BRDF), and presented the caveats of the connection of this
framework with measurements. Reflectance nomenclature in
the remote sensing and laboratory/field measurement literature
has loosely followed the framework of Nicodemus et al. (1977)
28
G. Schaepman-Strub et al. / Remote Sensing of Environment 103 (2006) 27–42
and yet the caveats remain largely ignored, leaving a literature
rife with loosely understood concepts and measured quantities
that cannot be compared or modeled adequately due to erroneous or ambiguous nomenclature. Whereas it was hypothesized (Koestler, 1959) and later disproved (Gingerich, 2004)
that Copernicus' de Revolutionibus (Copernicus, 1543) was not
thoroughly read and understood, past and current remote
sensing literature demonstrate that Nicodemus et al. (1977) has
in fact been widely misunderstood, misapplied, or ignored. In
this work, we extend the definition of reflectance nomenclature
to its application specifically in remote sensing and field measurements, and present the implications of terminology misuse
when inferring surface physical parameters. We consider a
physically consistent, explicit declaration of reflectance quantities to be a prerequisite in peer-reviewed literature.
The physically based terminology proposed by Nicodemus
et al. (1977) was updated and adapted to the remote sensing case
by Martonchik et al. (2000), and the concept of the BRDF was
then extended to horizontally heterogeneous surfaces by Snyder
(2002). It was further generalized by Di Girolamo (2003) to
include the area of illumination and the area of measurement as
dependent variables, and thus explaining experimental results
which questioned the validity of the reciprocity of the BRDF.
On the practical side, recent advances in data production by the
science team for the NASA Multi-angle Imaging SpectroRadiometer (MISR) have led to a more uniform and physically
consistent reflectance terminology for operational reflectance
products. The MISR data products that include different reflectance quantities allow users to infer appropriate physical quantities for their investigations.
Despite these advancements, the peer-reviewed literature
shows that the remote sensing community still often neglects
the description of the physical conditions of measurements and
the corresponding terminology of at-surface reflectance quantities. This problem comes in the form of ambiguous or erroneous usage of the terminology.
The most striking oversight by our community is that of the
“measurement of the BRDF” in which neither the actual irradiance nor measured reflected field is properly directional. As
Nicodemus et al. (1977) state, “The BRDF itself, as a ratio of
infinitesimals, is a derivative with “instantaneous” [in angle]
values that can never be measured directly.”
Under natural conditions for Earth observing, the assumption
of a single direction of the incident beam for all ground based,
airborne, and spaceborne optical sensor measurements is invalid.
Natural irradiance is composed of a direct component (nonscattered radiation) and a diffuse component scattered by the
atmosphere (gases, aerosols, and clouds), and the surroundings of
the observed surface. The magnitude and spectral distribution of
the diffuse irradiance therefore depends on atmospheric conditions, topography, and the reflectance properties of the topography. Without correction of the diffuse component of the
irradiance, observed reflectance quantities are influenced by the
above effects and do not represent the desired intrinsic directional
reflectance characteristics of the observed surface.
The analysis of illumination effects present in measured
reflectance data is discussed in the literature (Deering & Eck,
1987; Kriebel, 1976, 1978; Liu et al., 1994; Ranson et al.,
1985). Several studies have analyzed simulated reflectance data
under different ratios of direct and diffuse irradiance conditions
(e.g., Asrar & Myneni, 1993; Lewis & Barnsley, 1994).
Lyapustin and Privette (1999) showed that the reflectance
function derived from measurements performed under ambient
sky conditions contains an atmospheric contribution, and
therefore shows considerable shape distortions with respect to
the inherent surface BRDF. Thus, experimental research on the
inherent reflectance anisotropy of the observed surface requires
an accurate atmospheric correction as developed by Martonchik
(1994) and Lyapustin and Privette (1999).
Furthermore, the instantaneous field of view (IFOV) of the
instrument may integrate over a large viewing solid angle rather
than an infinitesimal differential solid angle specific to the
BRDF. Thus, remote sensing measurements do not properly
coincide with bidirectional reflectance quantities and resulting
products should only be considered as approximations to the
surface bidirectional reflectance, a fact often neglected.
Quantitative comparisons of reflectance products provided
by different airborne or satellite sensor systems with inconsistent definitions or data descriptions are difficult, inaccurate, or
impossible. An explicit description of irradiance conditions and
sensor geometric specifications (i.e. IFOV) of the delivered atsurface reflectance products is necessary for such comparisons.
This is true for discrete angular observations, as well as inferred
hemispherical products such as albedo.
Given the state of terminology currently in practice, this
paper first gives an extensive physical and mathematical
description of different reflectance quantities and separates
conceptual from measurable quantities. Based on this background, we present examples from peer-reviewed literature that
exhibit frequent ambiguous or wrong usage of current
reflectance terminology. We then systematically highlight
differences in at-surface reflectance quantities by their definition. We focus on the geometry of the solid angle of the
illumination, i.e., directional and hemispherical extent. We
quantitatively compare operational MISR reflectance products
with respect to the corresponding atmospheric optical depth and
resulting wavelength-specific differences for selected biomes
and then perform modeling studies for two significantly
different natural surfaces, forest and snow. Using a variation
of the direct to diffuse irradiance ratio in the corresponding
radiative transfer models (i.e. Rahman–Pinty–Verstraete
(RPV), Discrete Ordinate Radiative Transfer (DISORT)), we
obtain quantitative results of the wavelength-dependent influence of the diffuse component on the hemispherical–directional
surface reflectance, i.e., for an illumination of hemispherical
extent.
2. Definitions
In the following, we present the definitions of most
commonly used reflectance quantities, based on the initial
terminology of Nicodemus et al. (1977) and Martonchik et al.
(2000), using notations listed in Table 1, and present associated
examples of their implementation.
G. Schaepman-Strub et al. / Remote Sensing of Environment 103 (2006) 27–42
Table 1
Notations used for the definition of at-surface reflectance quantities
29
to the radiance L [W m− 2 sr− 1] of a Lambertian surface is a
factor of π (Palmer, 1999).
Symbols
S
A
Φ
E
L
M
ρ
R
θ
ϕ
ω
Ω
λ
f()
Distribution of direction of radiation
Surface area [m2]
Radiant flux [W]
Irradiance, incident flux density; ≡ dΦ/dA [W m− 2]
Radiance; ≡ d2Φ/(dA · cosθ · dω) [W m− 2 sr− 1]
Radiant exitance, exitent flux density; ≡ dΦ/dA [W m− 2]
Reflectance; ≡ dΦr/dΦi [dimensionless]
Reflectance factor; ≡ dΦr/dΦrid [dimensionless]
Zenith angle, in a spherical coordinate system [rad]
Azimuth angle, in a spherical coordinate system [rad]
Solid angle; ≡ ∫dω ≡ ∫∫sinθ · dθ · dϕ [sr]
Projected solid angle; ≡ ∫cosθ · dω ≡ ∫∫cosθ · sinθ · dθ · dϕ [sr]
Wavelength of radiation [nm]
Function
Sub- and superscripts
i
Incident
r
Reflected
id
Ideal (lossless) and diffuse (isotropic or Lambertian)
atm
Atmospheric
dir
Direct
diff
Diffuse
Terms
BHR
BRDF
BRF
DHR
HDRF
BiHemispherical Reflectance
Bidirectional Reflectance Distribution Function
Bidirectional Reflectance Factor
Directional–Hemispherical Reflectance
Hemispherical–Directional Reflectance Factor
2.1. Radiance, reflectance, reflectance factors
The prerequisite for physically based, quantitative analysis
of airborne and satellite sensor measurements in the optical
domain is their calibration to spectral radiance. The spectral
radiance is the radiant flux in a beam per unit wavelength and
per unit area and solid angle of that beam, and is expressed in
the SI units [W m− 2 sr− 1 nm− 1].
The ratio of the radiant exitance (M [W m− 2]) with the
irradiance (E [W m− 2]) results in the so-called reflectance.
Following the law of energy conservation, the value of the
reflectance is in the inclusive interval 0 to 1. The reflectance
factor is the ratio of the radiant flux reflected by a surface to that
reflected into the same reflected-beam geometry and wavelength range by an ideal (lossless) and diffuse (Lambertian)
standard surface, irradiated under the same conditions. For
measurement purposes, a Spectralon panel commonly approximates the ideal diffuse standard surface. Reflectance factors
can reach values beyond 1, especially for strongly forward
reflecting surfaces such as snow (Painter & Dozier, 2004). We
assume further that an isotropic behavior implies a spherical
source that radiates the same in all directions, i.e., the intensity
[W sr− 1] is the same in all directions, whereas the diffuse
(Lambertian) behavior refers to a flat reflective surface.
Consequently the intensity falls off as the cosine of the
observation angle with respect to the surface normal (Lambert's
law) and the radiance L [W m− 2 sr− 1] is independent of
direction. However, the ratio of the radiant exitance M [W m− 2]
2.2. Conceptual and measurable reflectance quantities
We symbolize reflectance and reflectance factor as
ρ(Si,Sr,λ) = reflectance, and
R(Si,Sr,λ) = reflectance factor,
where Si and Sr describe the angular distribution of all
incoming and reflected radiance observed by the sensor,
respectively. Si and Sr only describe a set of angles occurring
with the incoming and reflected radiation and not their intensity
distributions. Sr represents a cone with a given solid angle
corresponding to a sensor's instantaneous field of view (IFOV),
but no sensor weight functions are included here. If the
sensitivity of the sensor depends on the location within the rim
of the cone, a response function should be included. When a
sensor has a different IFOV for different wavelength ranges,
then Sr depends on the wavelength.
The terms Si and Sr can be expanded into a more explicit
angular notation to address the remote sensing problem:
ρ(θi,ϕi,ωi;θr,ϕr,ωr;λ), and
R(θi,ϕi,ωi;θr,ϕr,ωr;λ),
where the directions (θ and ϕ are the zenith and azimuth
angle, respectively) of the incoming (subscript i) and the
reflected (subscript r) radiance, and the associated solid angles
of the cones (ω) are indicated. This notation follows the
definition of a general cone.
From a physical point of view, we may define special cases
of ρ and R in terms of conceptual quantities and measurable
quantities. Conceptual quantities of reflectance include the
assumption that the size to distance ratio of the illumination
source (usually the sun or lamp) and the observing sensor is
zero. They are labeled directional in the general terminology.
Since infinitesimal elements of solid angle do not include
measurable amounts of radiant flux (Nicodemus et al., 1977),
and unlimited small light sources and sensor IFOVs do not
exist, all measurable quantities of reflectance are performed in
the conical or hemispherical domain of geometrical considerations. Thus, actual measurements always involve non-zero
intervals of direction and the underlying basic quantity for all
radiance and reflectance measurements is the conical case.
For surface radiance measurements made from space, aircraft
or on the ground, under ambient sky conditions, the incident
radiance cone is of hemispherical extent (ω = 2π [sr]). The
irradiance can then be divided into a direct sunlight component
and a second irradiance component, namely sunlight which has
been scattered by the atmosphere, the terrain, and surrounding
objects, resulting in an anisotropic, diffuse illumination.
Because the ratio of diffuse to direct irradiance is a function
of wavelength, it highly influences the spectral dependence of
directional effects as shown in the quantitative comparison
section of this paper.
30
G. Schaepman-Strub et al. / Remote Sensing of Environment 103 (2006) 27–42
The above reflectance and reflectance factor definitions lead
to the following special cases:
• ωi or ωr are omitted when either is zero (directional quantities).
• If 0 b (ωi or ωr) b 2π, then θ,ϕ describe the direction of the
center axis of the cone (e.g. the line from a sensor to the
center of its ground field of view—conical quantities).
• If ωi = 2π, the angles θi,ϕi indicate the direction of the
incoming direct radiation (e.g., the position of the sun). For
remote sensing applications, it is often useful to separate the
natural incoming radiation into a direct (neglecting the sun's
size) and hemispherical diffuse part. One may also include a
terrain reflected diffuse component that is calculated with a
topographic radiation model such as TOPORAD (Dozier,
1980). Consequently, the preferred notation for the geometry
of the incoming radiation under ambient illumination
conditions is θi,ϕi,2π. Note that in this case, θi,ϕi describe
the position of the sun and not the center of the cone (2π),
except if the sun's position is at nadir. In the case of an
isotropic diffuse irradiance field, without any direct
irradiance component (closest approximated in the case of
an optically thick cloud deck), θi,ϕi are omitted.
• If ωr = 2π, θr and ϕr are omitted.
Finally, according to Nicodemus et al. (1977), the angular
characteristics of the incoming radiance are named first in the
term, followed by the angular characteristics of the reflected
radiance. This leads to the attributes of radiance and reflectance
quantities as illustrated in Table 2.
It should be noted that the nine standard reflectance terms
defined by Nicodemus et al. (1977) “are applicable only to
situations with uniform and isotropic radiation throughout the
incident beam of radiation”. They then state that “If this is not
true, then one must refer to the more general expressions”. This
implies that any significant change to the nine reflectance
concepts when the incident radiance is anisotropic lies in the
mathematical expression used in their definition. Based on this
implication, Martonchik et al. (2000) adapted the terminology
and reflectance names to the remote sensing case, which
involves direct and diffuse sky illumination. In the following,
we give the mathematical description of the most commonly
used quantities in remote sensing, thus the general expressions
for non-isotropic incident radiation. When applicable, we
simplify the expression for the special case of isotropic incident
radiation. The nine possible combinations of beam geometries
of the incident and reflected radiant fluxes are indicated as
Cases 1 to 9, corresponding to the illustrations in Table 2.
2.2.1. The bidirectional reflectance distribution function
(BRDF)—Case 1
The bidirectional reflectance distribution function
(BRDF) describes the scattering of a parallel beam of incident
light from one direction in the hemisphere into another direction
in the hemisphere. The term BRDF was first used in the
literature in the early 1960s (Nicodemus, 1965). Being
expressed as the ratio of infinitesimal quantities, it cannot be
directly measured (Nicodemus et al., 1977). The BRDF
describes the intrinsic reflectance properties of a surface and
Table 2
Relation of incoming and reflected radiance terminology used to describe reflectance quantities
Incoming/Reflected
Directional
Conical
Hemispherical
Directional
Bidirectional
CASE 1
Directional–conical
CASE 2
Directional–hemispherical
CASE 3
Conical
Conical–directional
CASE 4
Biconical
CASE 5
Conical–hemispherical
CASE 6
Hemispherical
Hemispherical–directional
CASE 7
Hemispherical–conical
CASE 8
Bihemispherical
CASE 9
The labeling with ‘Case’ corresponds to the nomenclature of Nicodemus et al. (1977). Grey fields correspond to measurable quantities (Cases 5, 8), the others
(Cases 1–4, 6, 7, 9) denote conceptual quantities. Please refer to the text for the explanation on measurable and conceptual quantities.
G. Schaepman-Strub et al. / Remote Sensing of Environment 103 (2006) 27–42
thus facilitates the derivation of many other relevant quantities,
e.g., conical and hemispherical quantities, by integration over
corresponding finite solid angles.
The spectral BRDF can be expressed as
BRDFk ¼ fr ðhi ; /i ; hr ; /r ; kÞ
dLr ðhi ; /i ; hr ; /r ; kÞ −1 sr :
¼
dEi ðhi ; /i ; kÞ
ð1Þ
For reasons of clarity, we will omit the spectral dependence
in the following. We therefore write for the BRDF
BRDF ¼ fr ðhi ; /i ; hr ; /r Þ ¼
dLr ðhi ; /i ; hr ; /r Þ −1 sr :
dEi ðhi ; /i Þ
ð2Þ
2.2.2. Reflectance factors—definition of Case 1 and Case 7
When reflectance properties of a surface are measured, the
procedure usually follows the definition of a reflectance factor.
The reflectance factor is the ratio of the radiant flux reflected by
a sample surface to the radiant flux reflected into the identical
beam geometry by an ideal (lossless) and diffuse (Lambertian)
standard surface, irradiated under the same conditions as the
sample surface. Following the different beam geometries of the
incident and reflected radiant fluxes as mentioned above, we
define the bidirectional reflectance factor, the hemispherical–
directional reflectance factor, the biconical reflectance factor,
and the hemispherical–conical reflectance factor.
The bidirectional reflectance factor (BRF; Case 1) is
given by the ratio of the reflected radiant flux from the surface
area dA to the reflected radiant flux from an ideal and diffuse
surface of the same area dA under identical view geometry and
single direction illumination:
BRF ¼ Rðhi ; /i ; hr ; /r Þ ¼
dUr ðhi ; /i ; hr ; /r Þ
dUid
r ðhi ; /i Þ
ð3Þ
the quantity dependent on the actual, simulated or assumed
atmospheric conditions and the reflectance of the surrounding
terrain. This includes spectral effects introduced by the variation
of the diffuse to direct irradiance ratio with wavelength (e.g.,
Strub et al., 2003).
HDRF ¼ Rðhi ; /i ; 2p; hr ; /r Þ ¼
¼
dEi ðhi ; /i Þ dLr ðhi ; /i ; hr ; /r Þ
d
¼ id
dLr ðhi ; /i Þ
dEi ðhi ; /i Þ
¼
fr ðhi ; /i ; hr ; /r Þ
¼ pdfr ðhi ; /i ; hr ; /r Þ:
frid ðhi ; /i Þ
dUr ðhi ; /i ; 2p; hr ; /r Þ
dUid
r ðhi ; /i ; 2pÞ
coshr sinhr Lr ðhi ; /i ; 2p; hr ; /r Þdhr d/r dA
coshr sinhr Lid
r ðhi ; /i ; 2pÞdhr d/r dA
Lr ðhi ; /i ; 2p; hr ; /r Þ
¼
¼
Lid
r ðhi ; /i ; 2pÞ
R
2p frRðhi ; /i ; hr ; /r ÞdUi ðhi ; /i Þ
2p ð1=pÞdUi ðhi ; /i Þ
¼
fr ðhi ; /i ; hr ; /r Þcoshi sinhi Li ðhi ; /i Þdhi d/i
:
R 2p R p=2
ð1=pÞ 0 0 coshi sinhi Li ðhi ; /i Þdhi d/i
0
0
¼
fr ðh0 ; /0 ; hr ; /r ÞEdir ðh0 ; /0 Þ þ
An ideal Lambertian surface reflects the same radiance in all
view directions, and its BRDF is 1/π. Thus, the BRF [unitless]
of any surface can be expressed as its BRDF [sr− 1] times π (Eq.
(6)). For Φrid and Lrid , we omit the view zenith and azimuth
angles, because there is no angular dependence for the ideal
Lambertian surface.
The concept of the hemispherical–directional reflectance
factor (HDRF; Case 7) is similar to the definition of the BRF,
but includes irradiance from the entire hemisphere. This makes
ð9Þ
ð10Þ
R 2p R p=2
fr ðhi ; /i ; hr ; /r Þcoshi sinhi Ldiff
i ðhi ; /i Þdhi d/i
0
R 2p R p=2
diff
ð1=pÞðEdir ðh0 ; /0 Þ þ 0 0 coshi sinhi Li ðhi ; /i Þdhi d/i Þ
0
ð11Þ
Lidiff
then, if and only if
is isotropic (i.e. independent of the
angles), we may continue
¼
R 2p R p=2
fr ðh0 ; /0 ; hr ; /r ÞEdir ðh0 ; /0 Þ þ Ldiff
fr ðhi ; /i ; hr ; /r Þcoshi sinhi dhi d/i
i
0
0
h
i
R 2p R p=2
diff
ð1=pÞ Edir ðh0 ; /0 Þ þ Li
coshi sinhi dhi d/i
0
0
ð12Þ
ð1=pÞEdir ðh0 ; /0 Þ
¼ pfr ðh0 ; /0 ; hr ; /r Þ
ð1=pÞEdir ðh0 ; /0 Þ þ Ldiff
i
Z 2p Z p=2
þ
fr ðhi ; /i ; hr ; /r Þcoshi sinhi dhi d/i
0
Ldiff
i
ð1=pÞEdir ðh0 ; /0 Þ þ Ldiff
i
ð13Þ
¼ Rðh0 ; /0 ; hr ; /r Þd þ Rð2p; hr ; /r Þð1−dÞ;
ð6Þ
ð8Þ
If we divide Li into a direct (Edir with angles θ0,ϕ0) and diffuse
part, we may continue
ð4Þ
ð5Þ
ð7Þ
R 2p R p=2
0
coshr sinhr dLr ðhi ; /i ; hr ; /r Þdhr d/r dA
¼
coshr sinhr dLid
r ðhi ; /i Þdhr d/r dA
31
ð14Þ
where d corresponds to the fractional amount of direct radiant
flux (i.e. d ∈ [0,1]).
The biconical reflectance factor (conical–conical reflectance factor, CCRF; Case 5), is defined as
CCRF ¼ R
R ðhiR; /i ; xi ; hr ; /r ; xr Þ
fr ðhi ; /i ; hr ; /r ÞdLi ðhi ; /i ÞddXi ddXr
R
¼ xr xi
;
ðXr =pÞd xi Li ðhi ; /i ÞddXi
ð15Þ
where Ω = ∫dΩ = ∫cosθdω = ∫∫cosθsinθdθdϕ is the projected
solid angle of the cone.
Formally, the CCRF can be seen as the most general quantity,
because its expression contains all other cases as special ones:
for ω = 0 the integral collapses and we obtain the directional
32
G. Schaepman-Strub et al. / Remote Sensing of Environment 103 (2006) 27–42
case, and for ω = 2π we obtain the hemispherical case. However,
the BRF and BRDF remain the most fundamental and desired
quantities because they are the only quantities not integrated
over a range of angles.
For large IFOV sensor measurements performed under
ambient sky illumination, the assumption of a zero interval
of the solid angle for the measured reflected radiance beam
does not hold true. The resulting quantity most precisely
could be described as hemispherical–conical reflectance
factor (HCRF; Case 8), obtained from Eq. (15) by setting
ωi = 2π:
HCRF ¼ R
R ðhiR; /i ; 2p; hr ; /r ; xr Þ
fr ðhi ; /i ; hr ; /r ÞdLi ðhi ; /i ÞddXi ddXr
R
¼ xr 2p
:
ðXr =pÞd 2p Li ðhi ; /i ÞddXi
¼
dUi ðhi ; /i Þ
Z
2p
¼
Z
0
p=2
R 2p R p=2
0
0
fr ðhi ; /i ; hr ; /r Þcoshr sinhr dhr d/r
dUi ðhi ; /i Þ
fr ðhi ; /i ; hr ; /r Þcoshr sinhr dhr d/r :
ð16Þ
ð17Þ
ð18Þ
ð19Þ
0
The bihemispherical reflectance (BHR; Case 9), generally
called albedo, is the ratio of the radiant flux reflected from a unit
surface area into the whole hemisphere to the incident radiant
flux of hemispherical angular extent.
BHR ¼ qðhi ; /i ; 2p; 2pÞ ¼
¼
dA
dUr ðhi ; /i ; 2p; 2pÞ
dUi ðhi ; /i ; 2pÞ
ð20Þ
R 2p R p=2
0
dLr ðhi ; /i ; 2p; hr ; /r Þcoshr sinhr dhr d/r
R 2p R p=2
dA 0 0 dLi ðhi ; /i Þcoshi sinhi dhi d/i
0
0
0
0
0
fr ðhi ; /i ; hr ; /r Þcoshr sinhr dhr d/r Li ðhi ; /i Þcoshi sinhi dhi d/i
R 2p R p=2
Li ðhi ; /i Þcoshi sinhi dhi d/i
0
0
(22)
R 2p R p=2
¼
0
0
qðhi ; /i ; 2pÞLi ðhi ; /i Þcoshi sinhi dhi d/i
:
R 2p R p=2
L
ðh
;
/
Þcosh
sinh
dh
d/
i
i
i
i
i
i
i
0
0
ð23Þ
If as before we divide Li into a direct (Edir with angles θ0,ϕ0)
and diffuse part, and assume that Lidiff is isotropic we can write
¼
R 2p R p=2
qðh0 ; /0 ; 2pÞEdir ðh0 ; /0 Þ þ pLdiff
qðhi ; /i ; 2pÞcoshi sinhi dhi d/i
i ð1=pÞ 0
0
Edir ðh0 ; /0 Þ þ pLdiff
i
(24)
2.2.3. Reflectance—Case 3 and Case 9
In the following, we describe the hemispherical reflectance
as a function of different irradiance scenarios including (i) the
special condition of pure direct irradiance, (ii) common Earth
irradiance, composed of diffuse and direct components, and (iii)
pure diffuse irradiance.
The directional–hemispherical reflectance (DHR; Case
3) corresponds to pure direct illumination (reported as blacksky albedo in the MODIS product suite (Lucht et al., 2000)). It
is the ratio of the radiant flux for light reflected by a unit surface
area into the view hemisphere to the illumination radiant flux,
when the surface is illuminated with a parallel beam of light
from a single direction.
dUr ðhi ; /i ; 2pÞ
DHR ¼ qðhi ; /i ; 2pÞ ¼
dUi ðhi ; /i Þ
R 2p R p=2
dA 0 0 dLr ðhi ; /i ; hr ; /r Þcoshr sinhr dhr d/r
¼
dUi ðhi ; /i Þ
R 2p R p=2 R 2p R p=2
¼
ð21Þ
¼ qðh0 ; /0 ; 2pÞd þ qð2p; 2pÞð1−dÞ;
ð25Þ
where d again corresponds to the fractional amount of direct
radiant flux.
For the special case of pure diffuse isotropic incident
radiation, a situation that may be most closely approximated in
the field by a thick cloud or aerosol layer, the resulting BHR
(reported as white-sky albedo in the MODIS product suite
(Lucht et al., 2000)) can be described as follows
BHR ¼ qð2p; 2pÞ
Z Z
1 2p p=2
¼
qðhi ; /i ; 2pÞcoshi sinhi dhi d/i :
p 0
0
ð26Þ
Under ambient illumination conditions, the albedo is
influenced by the combined diffuse and direct irradiance. To
obtain an approximation of the albedo for ambient illumination
conditions (also reported as blue-sky albedo in the MODIS
product suite), it is suggested to linearly combine the BHR for
isotropic diffuse illumination conditions and the DHR (see Eq.
(25)), corresponding to the actual ratio of diffuse to direct
illumination (Lewis & Barnsley, 1994; Lucht et al., 2000). The
diffuse component then can be expressed as a function of
wavelength, optical depth, aerosol type, and terrain contribution.
The underlying assumption of an isotropic diffuse illumination
may lead to significant uncertainties due to ignoring the actual
distribution of the incoming diffuse radiation.
All abovementioned albedo values, with the exception of the
BHR for pure diffuse illumination conditions, depend on the
actual illumination angle of the direct component. Thus, it is
highly recommended to include the solar geometry when describing albedo quantities.
2.3. Examples for measurable quantities and derived products
In Table 2, we present the typical incoming and reflected
beam geometries in relation to the individual reflectance cases
discussed above. From a strict physical point of view, the most
common measurement setup of satellites, airborne and field
instruments corresponds to the hemispherical–conical configuration (Case 8) (e.g., MERIS/ENVISAT, Analytical Spectral
Devices FieldSpec). In general, space-based instruments with a
spatial resolution of about 1 km have a IFOV with a full cone
G. Schaepman-Strub et al. / Remote Sensing of Environment 103 (2006) 27–42
angle of approximately 0.1° (e.g., MISR, MODIS, AVHRR). If
the HDRF is constant over the full cone angle of the instrument
IFOV, then the HCRF numerically equals the HDRF. Based on
this assumption, the reflectance products from these instruments
are often being referred to as HDRF and BRF instead of HCRF
and DCRF, without further correction for the conical observation angle. Field instruments like PARABOLA or ASG with a
IFOV full cone angle of about 4° to 5°, the significance of the
surface directional reflectance variability needs to be investigated. As long as the variability is unknown, these measurements should be reported as HCRF.
Albedometer measurements approximate the bihemispherical configuration (Case 9) (e.g., Kipp & Zonen, 2000). Finally, a
typical laboratory setup corresponds to the biconical configuration (Case 5), where a light source illuminates a target that is
measured using a non-imaging spectroradiometer (e.g., EGO
(Koechler et al., 1994), LAGOS (Dangel et al., 2005), or ASG
(Painter et al., 2003) used in the laboratory). For a perfectly
collimated light source (find a detailed description of the problem in Dangel et al., 2005), these measurements approximate
the directional–conical quantity.
The derivation of different at-surface reflectance quantities
from measurements requires a sophisticated processing, as
implemented, for example, in the MISR scheme. The integration of the at-surface HDRF (Case 7) over the viewing
hemisphere results in the BHR (Case 9). Using a modeling
approach to eliminate the diffuse illumination effects of the
HDRFs (Lyapustin & Privette, 1999; Martonchik, 1994;
Martonchik et al., 1998), BRF data are derived (Case 1).
The hemispherical integration of the BRFs over the viewing
hemisphere yields DHR data. MISR albedo products include
BHR (blue-sky albedo) and DHR (black-sky albedo), whereas
MODIS albedo products include BHR for isotropic diffuse
illumination conditions (white-sky albedo) and DHR (blacksky albedo).
These derivations of conceptual reflectance quantities from
measured reflectance data include the application of a BRDF
model. Thus, derived conceptual quantities depend not only on
the sampling scheme, availability and accuracy of measured
data, but also on the properties and accuracy of the model.
3. Current usage of reflectance terminology
The multi-angular measurement configuration of MISR
facilitates the derivation of different reflectance products
using consistent terminology. For many other satellite and airborne systems, the user community is faced with products
simply called ‘surface reflectance’ or ‘apparent surface reflectance’. Without detailed information on the preprocessing of
the observed data and beam geometries of the resulting reflectance products, these data are subject to misinterpretation
and greater uncertainty.
We consider a physically consistent expression of reflectance
quantities as a prerequisite in peer-reviewed literature. The
presented concept of reflectance terminology is available to the
user community, yet the literature contains prolific ambiguous
and erroneous use of terminology for measured and derived
33
reflectance quantities. Below we present a selection of articles
as exemplary for a wide range of publications indicating the
breadth of the challenge.
3.1. Ambiguous usage
We identified a widely spread ambiguous use of the terms
‘reflectance’, ‘surface reflectance’, and ‘albedo’, as these do not
accurately specify measured or derived physical quantities. This
is of particular importance if details on data acquisition and
atmospheric correction are missing.
Amongst many others, Roberts et al. (1993) use the
ambiguous term ‘reflectance’ when referring to field measurements with a spectrometer and empirically corrected imaging
spectrometer data, and once mention that ‘bidirectional
reflectance’ is measured in the field. All measurements
discussed in the above reference are hemispherical–conical
reflectance factors due to the contribution of direct and diffuse
components to the irradiance field.
Most satellite sensor products are reported as ‘surface
reflectance’ (e.g., MODIS MOD09 (Vermote & Vermeulen,
1999), ETM+ data processed after Liang et al. (2001). When
data of different sensors are combined or compared, the physics
behind these products must be more carefully examined to
prevent ambiguous results in future studies (e.g., Fang et al.,
2004).
The ambiguous use of the terms ‘reflectance’ and ‘albedo’ is
especially critical when reference databases that contain different physical quantities are compiled. There is a high probability that the presented numbers will be applied without
analyzing uncertainties introduced by their different definitions,
especially when potential users are not familiar with remote
sensing data and processing (Breuer et al., 2003).
3.2. Erroneous usage
Secondly, the terms ‘directional’, ‘bidirectional’, and
‘BRDF’ are widely used when referring to measured reflectance
quantities. As explained above, these quantities cannot be
measured, but are approximated by measurements and subsequent atmospheric correction and angular modeling. Even
applying an atmospheric correction algorithm to measured
radiance data does not per se result in bidirectional reflectance
products, as different algorithms account for different parts of
the atmospheric contribution. We therefore briefly discuss the
different atmospheric contributions, as described in Lee and
Kaufman (1986).
The radiance of light reflected from the Earth and the atmosphere (Lr) is composed of three components:
diff
Lr ¼ Latm
þ Ldir
r
r þ Lr ;
ð27Þ
where Lratm is the radiance of light scattered from the direct sun
beam into the sensor's field of view (FOV) without being
reflected by the surface, and is often called atmospheric path
radiance. Lrdir is the radiance of sunlight transmitted directly
through the atmosphere, then reflected by the surface, and then
34
G. Schaepman-Strub et al. / Remote Sensing of Environment 103 (2006) 27–42
directly transmitted through the atmosphere. Diffuse radiance
Lrdiff consists of three components:
1
2
3
Ldiff
¼ Ldiff
þ Ldiff
þ Ldiff
;
r
r
r
r
ð28Þ
whereas Lrdiff1 is the contribution of diffuse light scattered by the
atmosphere before reaching the surface and then directly
transmitted to the sensor, Lrdiff2 is the contribution of diffuse
light transmitted directly to the surface and then scattered by the
atmosphere toward a sensor after being reflected by the surface,
and Lrdiff3 is the contribution of light scattered by the atmosphere
both before and after being reflected from the surface. This
formulation is applicable for a uniform surface, whereas for
high spatial resolution remote sensing data, the adjacency effect
must also be corrected. This description shows that for the
derivation of bidirectional reflectance factors, the atmospheric
path radiance and all three diffuse components must be
subtracted from the reflected radiance, whereas many atmospheric correction algorithms account only for the path radiance
Lratm , and diffusely transmitted contributions to the sensor (i.e.,
Lrdiff2 and Lrdiff3).
For measurements close to the ground surface, the atmospheric path radiance (Lratm), Lrdiff2, and Lrdiff3 are essentially all
zero. Thus, these measurements already represent surfaceleaving radiance. Bidirectional reflectance properties can be
derived from ground reflectance measurements by deducing the
contribution of the hemispherical diffuse irradiance from the
total reflectance signal (e.g., Lyapustin & Privette, 1999;
Martonchik, 1994). Most publications report uncorrected
surface reflectance (Lyapustin & Privette, 1999) but refer to
them as ‘BRDF’ or ‘BRF’, instead of hemispherical–conical
reflectance quantities. Nolin and Liang (2000) indicate that “[e]
mpirically, the BRDF of a surface is determined by discrete
sampling of the angular radiance at finite solid angles.” The
important correction here is that the BRDF is approximated but
not determined by discrete sampling of the angular radiance at
finite solid angles.
Erroneous terminology is observed for measurements of most
multiangular field instruments, including PARABOLA (Deering
et al., 1990), FIGOS (Sandmeier & Itten, 1999), and other
multiangular measurement setups (e.g., Chopping, 2000; Susaki
et al., 2004). Some authors acknowledge that BRDF or BRF
cannot be measured, but–based on unphysical statements–still
refer to their measurements as bidirectional. Others suggest
technical solutions in the field (e.g. measuring the contribution of
the diffuse irradiance by shading of the direct sunlight using a
screen (Peltoniemi et al., 2005)). The measured reflectance under
diffuse conditions is then deduced from the total reflectance. As
the strong forward scattering phase function of aerosols dictates
that most of the diffuse irradiance follows a downward path close
to the original solar path (Lyapustin & Privette, 1999), a
significant part of the diffuse contribution is shaded by the screen
as well. The shading geometry itself indicates that a conical
irradiance is left after deducing the diffuse component outside the
shading screen. Further, the remaining conical irradiance is still
composed of a diffuse and direct part, and resulting reflectance
quantities are therefore biconical. This may result in spectral
distortions of the derived biconical reflectance distribution
function compared to the surface BRDF, and therefore is still
solely an approximation of the BRDF.
For airborne and spaceborne sensors, perturbations to the
downwelling and upwelling radiation streams contribute to the
overall observed reflectance. The wide variety of atmospheric
correction algorithms applied to airborne sensor data accounts
for different combinations of atmospheric contributions deduced from the at-sensor radiance to calculate the so-called ‘atsurface reflectances’. One example is the ATCOR atmospheric
processing chain (Richter & Schlapfer, 2002), where at-sensor
radiance is corrected for the path radiance and adjacency effect,
but not for the hemispherical irradiance. The resulting
ambiguously named ‘surface reflectance’ thus corresponds to
HDRF data. The multiangular data products of air- and
spaceborne POLDER are currently reported as ‘BRDF’ data,
while they lack a full aerosol correction (e.g., Bacour & Breon,
2005; Grant et al., 2003). On the other hand, the atmospheric
correction applied to AirMISR data delivers the full suite of
reflectance products (HDRF, BRF, BHR, and DHR) as available
for its spaceborne counterpart MISR.
Radiative transfer models typically allow the incorporation of
a pure direct, pure diffuse, or combined direct and diffuse
irradiance. Thus, depending on the purpose of modeled reflectance, BRF data (only direct irradiance), or HDRF data (pure
diffuse or a combination of direct and diffuse irradiance) can be
modeled. When running the models over a wide wavelength
range (e.g., for the analysis of imaging spectrometer data), the
diffuse to direct irradiance ratio must be wavelength dependent,
due to spectral nature of Mie and Rayleigh scattering. Current
vegetation studies often assume this ratio to be wavelength
independent (e.g., Atzberger, 2004; Koetz et al., 2004), and thus
modeled reflectances are expected to be spectrally distorted
compared to measured data.
Even though the terms ‘bidirectional’ and ‘BRDF’ are part
of the established nomenclature for the reflectance products of
many in situ, air- and spaceborne sensors, while still including
the hemispherical irradiance contribution, this does not change
the physical inadequacy of this terminology and the consequent
questionable interpretation of the respective reflectance products. As the terms ‘surface reflectance’ and ‘at-surface reflectance’ do not allow for the determination of the irradiance
conditions of the data product, we advise to follow a more
precise terminology as given above, using HDRF, HCRF, BRF,
DCRF, DHR, and BHR. For general study and data descriptions we further suggest using more appropriate terms, such as
‘reflectance anisotropy’ or ‘multiangular’, instead of the
commonly applied terms ‘BRDF’, ‘directional’, and ‘bidirectional’, while preserving the latter for the true conceptual
physical quantities.
4. Quantitative comparison of reflectance quantities
The following case studies highlight differences of the
described reflectance quantities using (i) MISR data products
for several selected biome types, (ii) model simulations for a
vegetation canopy, and (iii) model simulations for snow cover.
G. Schaepman-Strub et al. / Remote Sensing of Environment 103 (2006) 27–42
Table 3
Overview of MISR data selected for the analysis of the land surface reflectance
products
Site
Acquisition SZ
date in
[°]
2001
Howland,
Maine,
USA
Railroad
Valley,
Nevada,
USA
Mongu,
Zambia
07/21
27.7 0.10 Mixed forests, deciduous broadleaf
forests, evergreen needleleaf forests
08/17
28.4 0.10 Open shrublands, grasslands, woody
savannas
07/11
44.6 0.05 Savannas, cropland/natural
vegetation mosaic, woody
savannas, grasslands
24.1 0.31 Open shrublands, grasslands,
41.4 0.11 savannas
19.6 0.36 Open shrublands, barren or sparsely
vegetated, grasslands
25.2 0.07 Croplands, mixed forest, water
36.9 0.19 bodies
24.5 0.24 Evergreen needleleaf forests,
24.0 0.12 croplands, cropland/natural
vegetation mosaic, woody
savannas, mixed forests
Banizoumbou, 10/04
Niger
12/23
Hombori,
07/05
Mali
Avignon,
07/12
France
08/29
Bordeaux,
05/30
France
07/01
AOD IGBP biome type
SZ corresponds to the scene-averaged solar zenith angle, whereas AOD is the
scene-averaged aerosol optical depth at 558 nm over all valid pixels.
The differences of hemispherical versus directional reflectance
and reflectance factors (i.e., BHR (Case 9) versus DHR (Case 3)
and HDRF (Case 7) versus BRF (Case 1)) are computed for the
visible to shortwave infrared wavelength range, and different
ratios of direct to diffuse illumination conditions.
The analysis of MISR data highlights differences in reflectance
quantities resulting from data, algorithm, and model uncertainties,
as well as different aerosol optical depth (AOD). Thus, it reveals
expected differences in real satellite data products, whereas the
two simulation studies emphasize the general features of
reflectance quantities for vegetation and snow.
35
4.1. Analysis of MISR surface reflectance data products
4.1.1. Methods and selected data sets
Various land surface reflectance products are available from
the MISR sensor, retrieved from observations of nine cameras
(center view directions of 0°, ± 26.1°, ± 45.6°, ± 60.0°, and
± 70.5°) in four spectral bands (446, 558, 672, and 867 nm)
(Diner et al., 1999). The top-of-atmosphere MISR radiances are
atmospherically corrected to produce HDRF and BHR data,
surface reflectance properties as would be measured at ground
level but at the MISR spatial resolution of 1.1 km. The HDRF
and BHR are then corrected for diffuse illumination effects, and
fitted to a three-parameter empirical BRF model, resulting in the
BRF and DHR (Martonchik et al., 1998).
We statistically analyzed the differences of directional and
hemispherical reflectance products, namely DHR versus BHR
and BRF versus HDRF, by their mean values, mean absolute
and relative difference, and correlation.
The ratio of diffuse to direct illumination is an increasing
function of the atmospheric optical depth. The AOD decreases
with wavelength due mainly to the decreasing influence of
Rayleigh scattering and, to a lesser extent in most cases,
aerosols. For example, at 1000 hP, Rayleigh scattering
contributes 0.236, 0.094, 0.044, and 0.016 to the atmospheric
optical depth in the blue, green, red, and near-infrared band,
respectively (Russell et al., 1993). Thus, we expected the largest
product difference in short wavelength ranges, with a maximum
in the blue band.
We selected 10 2001 MISR data sets that span different
biome types (Friedl et al., 2002) and correspond to data product
version 12 (Lewicki et al., 2003). Three sites were covered
twice under different atmospheric conditions and sun zenith
angles (Table 3). The reliability of the land surface reflectance
values depends on the AOD magnitude, as the contribution of
the surface to the overall signal decreases with increasing AOD.
Therefore, pixels with a large optical depth in the green spectral
band (greater 0.5) were excluded from calculations. All
Fig. 1. Relation between BHR and DHR exemplified by the green spectral band of the Howland scene.
36
G. Schaepman-Strub et al. / Remote Sensing of Environment 103 (2006) 27–42
Table 4
Comparison of BHR and DHR values for the selected MISR scenes
Site
SZ
[°]
AOD,
Mean BHR/Mean((BHR − DHR)/BHR) [%]
558 nm
446 nm, 558 nm,
672 nm, 867 nm,
blue
green
red
NIR
Howland
Railroad
Valley
Mongu
Banizoumbou
27.7
28.4
0.10
0.10
0.031/2.1 0.053/1.5
0.095/1.7 0.137/1.3
0.028/1.1 0.318/0.7
0.170/0.9 0.238/0.7
44.6
24.1
41.4
19.6
25.2
36.9
24.5
24.0
0.05
0.31
0.11
0.36
0.07
0.19
0.24
0.12
0.046/0.5
0.060/1.2
0.084/0.5
0.108/5.1
0.045/2.0
0.050/0.9
0.059/1.5
0.048/1.8
0.094/0.3
0.176/1.4
0.261/0.6
0.349/1.6
0.069/0.9
0.079/0.7
0.087/1.4
0.073/1.0
Hombori
Avignon
Bordeaux
0.078/0.3
0.126/1.5
0.160/0.5
0.232/2.5
0.075/1.4
0.081/0.9
0.097/2.0
0.078/1.5
0.246/0.6
0.357/1.3
0.376/0.6
0.412/1.2
0.307/0.8
0.286/0.8
0.320/1.2
0.304/0.9
The aerosol optical depth at 558 nm, averaged over all analyzed pixels, is
indicated, as well as BHR mean values and BHR to DHR differences, in relation
to the BHR.
quantities called ‘scene-averaged’ rely on this exclusion.
Additionally, the products, ‘HDRF uncertainty averaged over
all cameras’ and the ‘relative BHR uncertainty’ were analyzed.
We assume that these uncertainties also apply to the BRF and
DHR products (http://eosweb.larc.nasa.gov/PRODOCS/misr/
Quality_Summaries/L2_AS_Products_20030125.html).
4.1.2. Differences between BHR (Case 9) and DHR (Case 3)
MISR BHR and DHR products are highly correlated in all
spectral bands and scenes (0.84 b r2 b 1.0) (Fig. 1). The relative
scene-averaged difference between BHR and DHR reaches a
maximum of 2.7% of the BHR value (with the exception of the
blue band of the Hombori scene reaching 5.1%) for all four
spectral bands (Table 4). The lowest scene-averaged relative
BHR uncertainty is 5.6% for the NIR spectral band of the
Avignon (07/12) scene, whereas relative BHR uncertainty can
reach 20% and higher, with a maximum of 88% for the blue
spectral band of the Banizoumbou (10/04) scene. Thus,
differences between BHR and DHR are very small compared
to actual product uncertainties.
We expect a trend of decreasing differences between BHR and
DHR with increasing wavelength, wherein the blue band
reflectances should show the biggest relative differences. This
is confirmed by results of five scenes, whereas for the other five
cases, differences reach the same or even higher values in at least
one of the other bands. Note that the differences are a
combination of effects caused by the AOD, scattering effects of
the surroundings, and the sensitivity of the BRDF to the angle of
the incident radiance with wavelength. Thus, if the surface shows
almost Lambertian reflectance properties in the blue spectral
region, but a much higher anisotropy in the NIR spectral region,
then the HDRF approximates the BRF (see Eq. (10)) for the blue
spectral region, whereas the difference between the two
quantities can be much bigger in the NIR spectral region.
Differences between the BHR and DHR product can be
related to the actual aerosol optical depth in the green spectral
band (Fig. 2). This relationship has r2 = 0.29 for the blue band
and increases with increasing wavelengths, with a maximum for
the NIR region (r2 = 0.79). A strong negative correlation is
found between the relative BHR to DHR difference and the
solar zenith angle (r2 = 0.51, 0.88, 0.82, 0.51 for the blue, green,
red, NIR band, respectively) (Fig. 3). The biggest difference
between BHR and DHR occurs around solar noon.
Green (558nm)
Blue (446nm)
0.06
y = 0.0661x + 0.0062
R2 = 0.2815
0.05
Mean((BHR-DHR)/BHR)
Mean((BHR-DHR)/BHR)
0.06
0.04
0.03
0.02
0.01
0
y= 0.0423x + 0.0065
R2 = 0.4725
0.05
0.04
0.03
0.02
0.01
0
0
0.1
0.2
0.3
Mean AOD (green)
0.4
0
Red (672nm)
y = 0.032x + 0.0044
R2 = 0.6486
0.05
0.04
0.03
0.02
0.01
0
0
0.1
0.2
0.3
Mean AOD (green)
0.4
NIR (867nm)
0.06
Mean((BHR-DHR)/BHR)
Mean((BHR-DHR)/BHR)
0.06
0.1
0.2
0.3
mean AOD (green)
0.4
y = 0.0225x + 0.0049
R2 = 0.7916
0.05
0.04
0.03
0.02
0.01
0
0
0.1
0.2
0.3
Mean AOD (green)
0.4
Fig. 2. Relation of relative BHR_DHR differences for all MISR spectral bands with mean aerosol optical depth in the green spectral band. Data points correspond to
analyzed scenes.
G. Schaepman-Strub et al. / Remote Sensing of Environment 103 (2006) 27–42
Green (558nm)
y = -0.0011x + 0.0503
R2 = 0.5055
0.05
Mean((BHR-DHR)/BHR)
Mean((BHR-DHR)/BHR)
Blue (446nm)
0.06
0.04
0.03
0.02
0.01
0
0
0.06
y = -0.0007x + 0.0352
R2 = 0.8812
0.05
0.04
0.03
0.02
0.01
0
10
20
30
50
Mean solar zenith [˚]
0
60
10
0.05
20
30
40
Mean solar zenith [˚]
50
NIR (867nm)
y = -0.0005x + 0.0232
R2 = 0.8226
Mean((BHR-DHR)/BHR)
Mean((BHR-DHR)/BHR)
Red (672nm)
0.06
37
0.04
0.03
0.02
0.01
0.06
y = -0.0002x + 0.0153
2
R = 0.5067
0.05
0.04
0.03
0.02
0.01
0
0
0
10
20
30
50
60
Mean solar zenith [˚]
0
10
20
30
40
Mean solar zenith [˚]
50
Fig. 3. Relation of relative BHR-DHR differences in all four MISR spectral bands with mean solar zenith angle. Data points correspond analyzed scenes.
4.1.3. Differences between HDRF (Case 7) and BRF (Case 1)
HDRF and BRF are highly correlated (r2 N 0.98) throughout
most spectral bands and view angles of all scenes (Fig. 4).
Compared to the quantities integrated over the view hemisphere,
the relative differences of the single view angle reflectances are
larger and reach up to 14% of the HDRF value. Regarding the
viewing direction, there is a significant trend of decreased
HDRF–BRF differences for the afterward looking cameras, with
the Ba camera (view zenith 45.6°) showing the smallest
differences for most scenes and spectral bands. Only a weak
correlation (r2 = 0.27) is found between the green spectral band
AOD and the relative HDRF–BRF difference averaged over all
spectral bands and cameras for each scene (Fig. 5).
4.2. Vegetation canopy reflectance simulations using the RPV
model
4.2.1. Methods and data
Using the PARABOLA instrument (Deering & Leone,
1986), black spruce forest HCRF data were observed at eight
solar zenith angles (35.1°, 40.2°, 45.2°, 50.2°, 55.0°, 59.5°,
65.0°, 70.0°) (Deering et al., 1995). As this study is based on a
past field measurements, we have to assume that the surface
directional reflectance variability within the PARABOLA IFOV
can be neglected and subsequently report measurements as
HDRF. After applying a simple HDRF to BRF atmospheric
correction (Tanre et al., 1983), data of the red spectral band
Fig. 4. Relation between HDRF and BRF exemplified by the green spectral band of the MISR nadir looking camera (An) of the Howland scene.
G. Schaepman-Strub et al. / Remote Sensing of Environment 103 (2006) 27–42
Mean[(HDRF-BRF)/HDRF)]
0.035
y = 0.0259x + 0.0188
R2 = 0.2712
0.03
0.025
0.02
0.015
0.01
0.005
0
0
0.1
0.2
0.3
Mean AOD (green)
0.4
Fig. 5. Correlation between mean aerosol optical depth in the green spectral
band and the relative HDRF-BRF difference, averaged over each scene, all
spectral bands and cameras of the MISR sensor.
(650–670 nm) were fitted to the parametric Rahman–Pinty–
Verstraete (RPV) model (Engelsen et al., 1996; Rahman et al.,
1993). We used resulting fit parameters and the RPV model to
simulate different reflectance quantities of a black spruce
canopy under various illumination conditions. The model was
run for a solar zenith angle of 30° and increments of direct (d)
and diffuse (1 − d) irradiance of d = 1.0, 0.8, 0.6, 0.4, 0.2, 0.0.
The irradiance scenario d = 1.0 corresponds to BRF or DHR,
whereas all others represent HDRF or BHR.
HDRF data. With decreasing direct irradiance, the anisotropy is
smoothed. Finally, the hot spot disappears for a scenario based
on diffuse irradiance only. Concentrating on nadir view data, the
relative difference between the bidirectional and the hemispherical–directional reflectance factor can be significant, especially
for illumination zenith angles around solar noon, approaching
the hot spot configuration (Fig. 7). The general pattern of
increasing differences between HDRF and BRF with decreasing
solar zenith angle agree qualitatively with the empirical results
of the MISR data analysis (Fig. 3). Even though absolute
differences between single BRF and HDRF data are numerically small for the selected wavelength range and certain
geometries, it becomes obvious that BRDF functions can be
strongly distorted when derived from model fits based on
HDRF instead of BRF data.
Also as expected for vegetation, the DHR increases with
increasing illumination zenith (Kimes, 1983), whereas the BHR
4.2.2. Results
As expected for a vegetation canopy, backscattering is the
dominating reflectance feature for the black spruce BRF data,
with a pronounced hot spot at a view zenith of 30° (Figs. 6 and
7). Adding an isotropic diffuse irradiance component results in
HDRF (d=0.8)
0.03
0.025
0.02
HDRF (d = 0)
0.015
0.01
-60
-40
-20
0
20
40
60
(backward) view zenith (forward)
BRF (d = 1)
0.03
reflectance factor
BRF (d=1.0)
BRF (d = 1)
0.035
reflectance factor
38
0.0275
0.025
0.0225
HDRF (d = 0)
0.02
0.0175
HDRF (d=0.6)
HDRF (d=0.4)
0.0387
0
10
20
30
40
50
60
70
illumination zenith
0.028
DHR (d = 1)
HDRF (d=0.2)
HDRF (d=0.0)
0.0104
reflectance
0.026
0.024
BHR (d = 0)
0.022
0.02
0
10
20
30
40
50
60
70
illumination zenith
Fig. 6. Reflectance factors of a black spruce forest canopy at 650–670 nm as a
function of view angle. The direct illumination, at 30° zenith, is from the left.
The six plots illustrate the influence of the relative amount of direct illumination.
The top left image corresponds to pure direct irradiation, thus to BRF data, all
others to HDRF data (from d = 0.8 to d = 0). The bottom right image corresponds
to totally diffuse illumination (d = 0).
Fig. 7. Simulated BRF (d = 1.0) and HDRF (d = 0.8 to d = 0.0) data for a black
spruce canopy in the solar principal plane (top); BRF (d = 1.0) and HDRF
(isotropic diffuse illumination conditions (d = 0.0)) at nadir view geometry
(centre); DHR (d = 1.0) and BHR (isotropic diffuse illumination (d = 0.0))
(bottom).
G. Schaepman-Strub et al. / Remote Sensing of Environment 103 (2006) 27–42
for isotropic diffuse irradiance conditions (white-sky albedo) is
independent of the illumination angle (Fig. 7). According to Eq.
(25), the albedo for ambient sky conditions (blue-sky albedo)
can be approximated by a combination of DHR and BHR for
isotropic incident radiation (white-sky albedo). For a given
illumination zenith angle (e.g., 20° (Fig. 8)) the approximated
BHR for ambient sky conditions then lies on a vertical line
between the DHR and BHR for isotropic incident radiation
which depend on the actual ratio of diffuse to direct irradiance.
39
the model are the single-scattering albedo, extinction efficiency,
and the single-scattering phase function, determined with a raytracing approach for spheroidal particles (Macke & Mishchenko,
1996). Displayed simulation results correspond to a spheroid of
minimum and maximum radii of 208 μm and 520 μm,
respectively. The multiple scattering model was run for a solar
zenith angle of 30° and increments of direct (d) and diffuse
irradiance of d = 1.0, 0.8, 0.6, 0.4, 0.2, and 0.0. These irradiance
scenarios correspond to BRF and DHR for pure direct illumination
conditions (d = 1.0), and HDRF and BHR for the remaining.
4.3. Snow reflectance simulations
4.3.1. Methods and data
This case study presents results from snow directional
reflectance simulations, coupling single-scattering parameters
and the discrete-ordinates multiple scattering model DISORT
(Stamnes et al., 1988). The single-scattering parameters used in
4.3.2. Results
In Fig. 8, we show the angular distributions of the irradiance
scenarios for wavelength 0.55 μm. The models for d = 1.0
through d = 0.2 exhibit a forward reflectance distribution that
decreases in magnitude with increasing diffuse component. For
the totally diffuse irradiance scenario, the distribution shows a
Fig. 8. Angular distribution of reflectance for a range of irradiance cases at 0.55 μm and solar zenith angle = 30°. The target center represents the nadir view geometry
θr, ϕr = (0°, 0°), radial distance from center represents the view zenith angle, and the angle about the center represents the view azimuth angle. The forward reflectance
direction is ϕr = 0°.
G. Schaepman-Strub et al. / Remote Sensing of Environment 103 (2006) 27–42
0.55 µm
1.00
DHR
Reflectance
shallow bowl shape. This minimum at nadir results from the
angular intersection of the strong forward scattering phase
function with the surface. The solar principal plane for these
scenarios is given in Fig. 9 (top).
The differences at nadir between the BRF and HDRF at
d = 0.8 and HDRF at d = 0.6 for λ = 0.55 μm are 0.02 and 0.04,
respectively, with maximum differences at view zenith angles of
10° to 20° in the forward direction. The bowl-shaped
distribution for diffuse irradiance becomes relatively deeper at
longer wavelengths (Fig. 9 (bottom)), such as 1.03 μm. We
show the 1.03 μm model because this is the wavelength range in
which snow reflectance is most sensitive to grain size and thus
is used in imaging spectroscopy models for inference of grain
size and albedo (Green et al., 2002; Nolin & Dozier, 2000). The
enhancement of the bowl shape at greater diffuse irradiance is
explained as above coupled with a decrease in the singlescattering albedo at the longer wavelengths. Only for the BRF
and d = 0.8 irradiance cases is the distribution properly forward
reflecting.
Fig. 10 shows DHR of snow relative to the illumination
zenith angle with the associated BHR for isotropic diffuse
illumination conditions. For both wavelengths, the DHR
increases with increasing zenith angle but the increase is far
greater in absolute and relative reflectance at 1.03 μm.
BHR
0.98
0.96
0.94
0
20
40
60
80
Illumination Zenith Angle
1.03 µm
0.70
DHR
Reflectance
40
BHR
0.60
0.50
0.40
0
5. Conclusions
20
40
60
80
Illumination Zenith Angle
The remote sensing community devotes major effort to
calibration and validation of sensors, measurements, and de0.55 µm
Reflectance Factor
1.10
1.00
BRF (d = 1.0)
HDRF (d = 0.8)
HDRF (d = 0.6)
HDRF (d = 0.4)
HDRF (d = 0.2)
HDRF (d = 0.0)
0.90
0.80
0.70 backward
60
40
forward
20
0
20
40
60
view zenith angle in principal plane
1.03 µm
Reflectance Factor
0.60
0.50
0.40
BRF (d = 1.0)
HDRF (d = 0.8)
HDRF (d = 0.6)
HDRF (d = 0.4)
0.30
HDRF (d = 0.2)
HDRF (d = 0.0)
forward
0.20 backward
60
40
20
0
20
40
60
view zenith angle in principal plane
Fig. 9. Simulated BRF (d = 1.0) and HDRF (d = 0.8 to d = 0.0) data for snow in
the solar principal plane at wavelengths 0.55 μm (top) and 1.03 μm (bottom).
Fig. 10. Directional_hemispherical reflectance versus illumination zenith angle
for snow at wavelengths 0.55 μm and 1.03 μm. The bihemispherical reflectance
for isotropic diffuse illumination (d = 0) is included for comparison.
rived products to quantify and reduce their uncertainties. Given
recent advances in instrument design, radiometric calibration,
atmospheric correction, algorithm development, product development, validation, and delivery, the lack of standardization of
reflectance terminology and products becomes a considerable
source of error.
The variety in physical quantities resulting from different
sensor sampling schemes, preprocessing, atmospheric correction, and angular modeling, requires a documentation standard
for remotely sensed reflectance data. Beyond the algorithm
theoretical basis document with a detailed description of the
data processing, a short and standardized description on the
physical character of the delivered reflectance products must be
accessible by the user. This necessarily includes the accurate
listing of opening angles and directions of illumination and
observation used, revealing whether the product represents
inherent reflectance properties of the surface or contains a
diffuse illumination component corresponding to the atmospheric and terrain conditions of the observations.
Relying on this standardized reflectance description, users
can then choose the reflectance products appropriate to their
applications, depending on the coupling of the reflectance with
the sky radiation. Inherent reflectance property data is available
from few sensors only, but as shown by Pinty et al. (2005),
crucial in, for example, climate modeling. Biases introduced by
using an inappropriate reflectance quantity can exceed
G. Schaepman-Strub et al. / Remote Sensing of Environment 103 (2006) 27–42
minimum sensitivity levels of climate models (i.e., ± 0.02
reflectance units (Sellers et al., 1995)), as shown by the
comparison of different MISR reflectance products in this study.
Further, they introduce systematic, wavelength dependent errors
in reflectance and higher level product validation efforts, in data
fusion approaches based on different sensors, and applications,
where so-called ‘a priori BRDF knowledge’ is applied (Li et al.,
2001). These differences are especially important in long-term,
large area trend studies, as the latter are mostly based on
multiple sensors with different angular sampling and modeling,
as well as atmospheric correction schemes.
The selected examples from the peer-reviewed literature
clearly show that the current use of reflectance terminology
often does not comply with physical standards and has implications on scientific results. In order to stimulate users to choose
the appropriate sensor and reflectance quantity and name it
accordingly, they must have access to an easy to understand,
physically based explanation of the delivered reflectance
product.
After several decades of custom and practice using a vague
reflectance terminology, it is important not to underestimate the
efforts of standardization. Semantic interoperability and standardization for the benefit of the increasing base of remote
sensing product customers has been recognized by various
bodies and initiatives (e.g., ISPRS, GEO, GEOSS, INSPIRE,
etc.), however the implementation will still take some time.
With this paper, we contribute to the awareness and emphasize
the importance of an ongoing discussion on reflectance
terminology, trying to avoid multiple redevelopment of similar
terminologies that would lead to a segregation of user
communities. The same scrutiny and transparency applied to
the peer-reviewed publication process should be applicable to
reflectance data product providers, resulting in open standards
facilitating data exchange and minimizing product
uncertainties.
Future research should emphasize an integrated standardization of remote sensing data products, including the calibration
process, the terminology of radiometric and photometric
quantities used, as well as proper linkage of these products
(e.g., various albedos) to physical models.
Acknowledgement
The work of G. Schaepman-Strub was supported by the
Swiss National Science Foundation (Project No. 200020101517 and PBZH2-108485) and a European Space Agency's
External Fellowship. The MISR data were obtained from the
NASA Langley Research Center Atmospheric Sciences Data
Center. We thank Michel Verstraete for the discussion on an
earlier version of the manuscript, and reviewers for their substantial comments.
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